Problem 103 - College Algebra

Question

Why is \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) not simplified? What must be done to simplify the expression?

Step-by-Step Solution

Answer

The simplified form of the expression is \(-6x^{-3}\).

Step 1: Identify the bases and their powers.

In this expression, both terms have a variable \(x\) which is our base. The base \(x\) in the first term is raised to the power of 2, and in the second term, it is raised to the power of -5.

Step 2: Apply the law of exponents.

The law of exponents states that when multiplying powers with the same base, we add the exponents. So, for our base \(x\) with the powers of 2 and -5, \(x^{2} \cdot x^{-5} = x^{2-5}\).

Step 3: Carry out the subtraction in the exponent.

Now perform the subtraction operation in the exponent as indicated. This gives: \(x^{2-5} = x^{-3}\).

Step 4: Multiply the numerical constants.

The numerical constants in our terms are -3 and 2. Multiply these constants together to give \(-3 \cdot 2 = -6\).

Step 5: Write the simplified expression.

Combine the result from steps 3 and 4 to form a single term. So, the simplified expression is \(-6x^{-3}\).